Duality and DeepMartingale for High-Dimensional Optimal Switching: Computable Upper Bounds and Approximation-Expressivity Guarantees
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910116023894016 |
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| author | Ye, Junyan Wong, Hoi Ying |
| author_facet | Ye, Junyan Wong, Hoi Ying |
| contents | We study finite-horizon optimal switching with discrete intervention dates on a general filtration, allowing continuous-time observations between decision dates, and develop a deep-learning-based dual framework with computable upper bounds. We first derive a dual representation for multiple switching by introducing a family of martingale penalties. The minimal penalty is characterized by the Doob martingales of the continuation values, which yields a fully computable upper bound. We then extend DeepMartingale from optimal stopping to optimal switching and establish convergence under both the upper-bound loss and an $L^2$-surrogate loss. We also provide an expressivity analysis: under the stated structural assumptions, for any target accuracy $\varepsilon>0$, there exist neural networks of size at most $c d^{q}\varepsilon^{-r}$ whose induced dual upper bound approximates the true value within $\varepsilon$, where $c$, $q$, and $r$ are independent of $d$ and $\varepsilon$. Hence, the dual solver avoids the curse of dimensionality under the stated structural assumptions. For numerical assessment, we additionally implement a deep policy-based approach to produce feasible lower bounds and empirical upper--lower gaps. Numerical experiments on Brownian and Brownian--Poisson models demonstrate small upper--lower gaps and favorable performance in high dimensions. The learned dual martingale also yields a practical delta-hedging strategy. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_08080 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Duality and DeepMartingale for High-Dimensional Optimal Switching: Computable Upper Bounds and Approximation-Expressivity Guarantees Ye, Junyan Wong, Hoi Ying Optimization and Control Numerical Analysis Probability 93E20, 68T07, 65Y20, 60G40 We study finite-horizon optimal switching with discrete intervention dates on a general filtration, allowing continuous-time observations between decision dates, and develop a deep-learning-based dual framework with computable upper bounds. We first derive a dual representation for multiple switching by introducing a family of martingale penalties. The minimal penalty is characterized by the Doob martingales of the continuation values, which yields a fully computable upper bound. We then extend DeepMartingale from optimal stopping to optimal switching and establish convergence under both the upper-bound loss and an $L^2$-surrogate loss. We also provide an expressivity analysis: under the stated structural assumptions, for any target accuracy $\varepsilon>0$, there exist neural networks of size at most $c d^{q}\varepsilon^{-r}$ whose induced dual upper bound approximates the true value within $\varepsilon$, where $c$, $q$, and $r$ are independent of $d$ and $\varepsilon$. Hence, the dual solver avoids the curse of dimensionality under the stated structural assumptions. For numerical assessment, we additionally implement a deep policy-based approach to produce feasible lower bounds and empirical upper--lower gaps. Numerical experiments on Brownian and Brownian--Poisson models demonstrate small upper--lower gaps and favorable performance in high dimensions. The learned dual martingale also yields a practical delta-hedging strategy. |
| title | Duality and DeepMartingale for High-Dimensional Optimal Switching: Computable Upper Bounds and Approximation-Expressivity Guarantees |
| topic | Optimization and Control Numerical Analysis Probability 93E20, 68T07, 65Y20, 60G40 |
| url | https://arxiv.org/abs/2604.08080 |